Article
A Unified Global Reference Frame of Vertical Crustal
Movements by Satellite Laser Ranging
Xinhui Zhu 1,2,*, Ren Wang 1, Fuping Sun 1 and Jinling Wang
2
Zhengzhou Institute of Surveying and M apping, 66 Longhai Road, Zhengzhou 450052, China;
[email protected] (R.W.); [email protected] (F.S.)
2 School of Civil and Environment Engineering, University of New South Wales, Sydney 2052, Australia;
[email protected] du.au
* Corresponde nce: [email protected]; Te l.: +61-488-031-190
1
Academic Editor: Radislav A. Potyrailo
Rece ive d: 5 De cember 2015; Acce pte d: 3 February 2016; Publishe d: 8 February 2016
Abstract: Crustal movement is one of the main factors influencing the change of the Earth system,
especially in its vertical direction, which affects people’s daily life through the frequent occurrence
of earthquakes, geological disasters, and so on. In order to get a better study and application of the
vertical crustal movement,as well as its changes, the foundation and prerequisite areto devise and
establish its reference frame; especially, a unified global reference frame is required. Since SLR
(satellite laser ranging) is one of the most accurate space techniques for monitoring geocentric
motion and can directly measure the ground station’s geocentric coordinates and velocities relative
to the centre of the Earth’s mass, we proposed to take the vertical velocity of the SLR technique in
the ITRF2008 framework as the reference frame of vertical crustal motion, which we defined as the
SLR vertical reference frame (SVRF). The systematic bias between other velocity fields and the
SVRF was resolved by using the GPS (Global Positioning System) and VLBI (very long baseline
interferometry) velocity observations, and the unity of other velocity fields and SVRF was
realized,as well. The results show that it is feasible and suitable to take the SVRF as a reference
frame, which has both geophysical meanings and geodetic observations, so we recommend taking
the SLR vertical velocity under ITRF2008 as the global reference frame of vertical crustal movement .
Keywords: vertical crustal movement; reference frame; SLR (satellite laser ranging); ITRF
(International Terrestrial Reference Frame); systematic bias model
1. Introduction
Along with the high frequency occurrence of earthquakes and geological disasters, it is
necessary to study crustal movements, especially in the vertical direction. Many experts have
researched the vertical crustal movements in different areas and regions [1–4]. For instance, Brown
and Oliver [1] studied the vertical crustal movements in the Eastern United States by levelling data
of the Vertical Control Net carried out by the NGS (National Geodetic Survey). With the
development of space geodetic techniques, people study vertical crustal movements and
earthquakes through very long baseline interferometry(VLBI), SLR and GPS, sometimes integrating
other geophysical methods [5–8]. Considerable effort and progress have also been made in the
evolution of vertical crustal deformation models [9–13]. For example, Bevis and Brown [13] sketched
the development of station trajectory models used in crustal movement and described some recent
generalizations of these models, which gave us a clear understanding in geodesy and geophysics.
However, the foundation and prerequisite for studying global vertical crus tal movement is actually
to establish its reference frame. Some scholars and data analysis centres have made some
exploratory and confirmatory studies on the reference frame of vertical crustal motion.
Mother et al. [14] pointed out that the geodetic refer ence system in four dimensions cannot be
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ignored because of the factors of Earth tides, plate motions, vertical crustal motions, and so on.
Yurkina [15] indicated that some difficulties existed when establishing a reference system in vertical
and crustal movement studies because of sea level changes considering variations of the Earth’s
gravitational field and displacements of the Earth’s centre of mass (CM), and he also described the
possibility by means of SLR techniques to study the vertical reference. Sun et al. [16] carried out a
preliminary study of the reference frame of vertical crustal movements by using a postglacial
rebound model. Dong et al. [17] indicated that an isostatic datum was inapplicable forstudying the
present vertical crustal movement of the Chinese continent; therefore, a single-point dynamic
vertical datum was discussed and established by using national high -precision levelling data.
Zhang et al. [3,11,12] studied the datum of regional vertical crustal movement, which is based on the
hypothesis of isostatic theory, and also analysed and fitted the deformation field of vertical crustal
motion. As this method did not adopt global data, it was still a regional datum. Moreover, the results
of global tectonic change proved that the hypothesis of an isostatic datum was not very
accurate [2,18,19]. Jekeli and Dumrongchai [20] constructed the regional vertical datum IGLD85
(International Great Lakes Datum of 1985) by using satellite altimetry and level data. Buraet al. [21]
realized a global vertical reference frame (GVRF) by using four regional and local vertical datums
(LV Ds) distributed worldwide through adopting the reference geopotential value, which was
considered as a demonstration that the methodology proposed can be applied ata global scale. Amos
and Featherstone [22] stated that New Zealand uses 13 separate local vertical datums based on
geodetic levelling from 12 different tidegauges, and they attempted to use iterated quasi -geoid
models to unify the vertical datum. Bevis et al. [23] presented a method for constructing and assessing
the stability of a geometrical reference frame of vertical crustal movement. Donnelly et al. [24] and
Haasdyket al. [25] focused on the necessity and realization of a new geodetic datum for Australia
and showed that if the geodetic datum is not updated, the systematic vertical bias will be 9 cm based
on different datums. Crespiet al. [26] reviewed the definition and realization of geodetic reference
frames and presented some relevant examples at both global and local levels, but did not expand the
analysis of vertical reference frames, although which are widely applied.
From the overall perspective of the Earth system, some definitions and conceptions, for
example the GVRF and the global vertical datum (GVD) [27–29], have been put forward by several
important organizations and symposiums, such as the IUGG (International Union of Geodesy and
Geophysics), IAG (International Associat ion of Geodesy), GGOS (Global Geodetic Observing
System), IGGOS (Integrated GGOS), and so on. The above realizations are all complex with
geometrically and geophysically-influenced factors and data collections, including the usual
physical components (geoid and physical heights) and geom etrical components (level ellipsoid and
ellipsoidal heights). Vertical crustal movement is part of the change of the whole Earth system, so in
this paper, we focus on the topic of the reference frame of vertical crustal movements, and we prefer
to discuss it in ageometrical way and also include geophysical information based on the SLR
technique itself.
A unified global reference frame of vertical crustal movements has a very significant meaning
for a more rational and accurate global coordinate reference frame. Because of the disunity, the
vertical velocity fields of global crustal movements are always distorted, even at the same
co-located sites, and different techniques and data processing software applications give different
results. Theoretically, vertical crustal movements should be defined as movement s relative to the
Earth’s CM. Th e vertical component of velocities in the same station should be the same. Actually,
data processing based on different reference frames brings different results ,as well. For the same
station’s vertical component, different space geodetic data analysis centres around the world also
publish different forms. For these reasons, the vertical data of various countries and regions in the
world havebeen difficult to unify for a long time in the past, and such a situation will not only
result in misunderstanding and confusion regarding application in the field of geodesy, but also
restrict the observations used in various seismic, geological and geophysical fields and may even be
misleading. Therefore, we definitely need to establish a unified global reference frame of vertical
crustal movements in order to unify their velocity fields.
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Velocity fields of global stations presented by different data analysis centres are almost
constrained by the velocity vector based on the absolute plate motion model NNR -NUVEL1A (No
net rotation-NUVEL1A)and established with one or more reference stations, while NNR-NUVEL1A
can only give the station’s horizontal direction velocity [30,31], which indicates that its implicit
constraint is that the vertical velocity of the reference station may be constrained aszero; for example,
Watkins and Eanes [32] constrained the Kauai site’s vertical velocity to be zero according to the
NNR-NUVEL1A model, which would have yielded vertical velocity uncertainties greater than 3
mm/yr. The actual situation is as follows: (1) the m ovement in the vertical direction of those
reference stations changes by varying degrees; (2) the reference stations selec ted by different data
analysis and processing centres are not exactly the same. The above conditions have caused disunity
and the inaccurate use of the global vertical crustal movement reference frame.
From the theoretical analysis, crustal vertical movements should be defined as a motion relative
to the Earth’s CM, viz.the centre-of-mass of the total Earth system, which is the solid Earth and its
fluid envelope and usually referred to as the geocenter [33]. The vertical velocity of one station
should be the same, even when it is obtained by using different techniques , but in fact, due to the
disunity of the reference frame of crustal vertical movements in different regions of the world, the
results from some of the data analysis centres and regional data processing were different, and the
resulting rate values differ even at the same station when they are obtained using different
observation techniques, which limits the application of space observation techniques in national
economical constructions. Therefore, it is necessary to unify the velocity fields of different data
analysis centres and establish a uniform global reference frame of vertical crustal movements, which
is currently an urgent problem in the field of global tectonics. This paper contributes by proposing a
method of establishing the global reference frame of vertical crusta l movements by using the SLR
technique, which has a clear geophysical significance, is easy to implement and is suitable as a
unified global reference frame of vertical crustal movements.
2.Method
2.1. Feasibility Analysis
After approximately 50 years of development, SLR is one of the most accurate space geodetic
techniques: its single measurement precision has reached the sub-centimetre level, and it is currently
the most accurate satellite tracking technique for monitoring the motion of the Earth’s CM [33,34].
SLR contributes to certain parameters of the reference frame, such a s the origin and the scale [35,36],
and the origin of the ITRF currently realized using only SLR data [33]. The ground station’s
coordinates with respect to the Earth’s CM can be directly observed by the SLR technique; therefore,
the station’s vertical velocity measured by SLR is also relative to the Earth’s CM. Since the
movement of the Earth’s CM reflects the complexity of movement and interaction within the Earth's
interior and various spheres, the vertical movement relative to the Earth’s CM can be seen as a
reference frame of vertical crustal movement.
If the Earth’s CM is taken as a reference point, its motion should be considered as relative to the
Earth’s centre. A geocenter refers to the CM of the entire Earth (including the oceans and
atmosphere), which is unique. There are two motions of the Earth’s CM: one is the movement of the
CM in space, such as the revolution of the Earth’s CM around the Sun; the other is in the Earth’s
interior, which is caused by the change in the distribution of the Earth’s internal mass. The whole
Earth (including the oceans and atmosphere) is a closed conservative system, because of the
conservation of momentum, and thus, in theory, changes in the distribution of the Earth’s internal
mass will not affect the motion of the Earth’s CM in space, but will only lead to variations of the
geocentric position in the solid Earth’s interior. Currently , the hot research topic is intended to mean
the latter geocentric movement.
Geocentric movement is a very complex process, including not only both short-term and
short-period items,but also long-term or long-term periods. Sun et al. [37], based on the observations
at the end of 19th centuryand theoretical calculations, indicated that at the time scale of 30 days to
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10 years, the movement of the geocenter is within 1 cm;its primary cycle is annual, while its
secondary one is six monthly, mainly due to the atmosphere, oceans and seasonal changes of surface
water distribution. In accordance with the rule of movement of the Earth’s CM, an average
geocentric position resolved in its primary cycle is called the conventional geocenter. A variety of
short-term or periodic-term changes of the instantaneous geocenter relative to the conventional
geocenter could be corrected by solving the stations ’ mean coordinates, w hile long-term or
long-cycle changes are reflected by the change of the station’s mean coordinates. Guo et al. [38]
observed the distance of satellite LAGEOS 1/2 (LAserGEOdynamics Satellite) by using SLR,
calculated the geocentric motion in the period 1993–2006, carried out an analysis and also found that
a long and periodic change in the geocentric motion exists, where seasonal variation is the primary
item, mainly through the mass distribution of the Earth fluid sphere, such as the oceans, atmosphere,
land water, and so on. Thus, the change of geocentric movement is mainly reflected in a short -term
or short-term period, while the long-term or long-term period did not change significantly,
moreover when resolved, which can be reflected in the change of the s tation’s mean coordinates. In
summary, when studying the station’s change in thelongterm or a long period, the geocentric centre
can be used as a reference point; namely, it is feasible to take the SLR vertical movement relative to
the centre of the Earth as a reference frame of vertical crustal motion.
ITRF2008 was released in 2010. The time span of observations adopted with the SLR technique
is 1983.0–2009.0, and the observation epoch isat 2005.0. The error of the observation stations’
geocentric coordinate velocity is mostly within 1 mm/yr [6]. The velocity field obtained by the SLR
technique under ITRF2008 is still based on time series, while compared toITRF2005, its observation
accuracy and numbers have greatly improved. In summary, it is feasible to ta ke the vertical velocity
field obtained by the SLR technique under ITRF2008 as a global reference frame of vertical crustal
movement. We discuss this in detail below and establish such a global unified reference frame of
vertical crustal movements.
2.2. Establ ish ment of Model
In theory, observation stations along the vertical direction of movement should be defined
relative to the movement of the Earth’s CM, but in practical application, each data analysis centre
has a certain arbitrariness in defining its vertical movement’s reference frame, mostly taking one or
more stations’ vertical velocity located in a tectonically-stable region constrained to zero as the
defining of the vertical reference frame, and therefore, the vertical movement after the reunification
in the concept is still not relative to the centre of the Earth. The SLR technique can directly measure
the geocentric coordinate and velocity field if a large number of SLR stations within high precisions
are selected on a global scale, whose measured vertical velocities constitute the reference frame of
the global vertical crustal movements. If the vertical velocity measured by SLR is relative to the
Earth’s CM, a systematic bias between the other space-based observations and the SLR’s
observations will inevitably exist. Therefore, in order to eliminate the systematic bias between the
SLR measured velocity field and the other counterparts ,it is possible to achieve a unified global
reference frame of vertical crustal movements relative to the Earth’s CM. The systematic bias of
different reference frames realized by different observed techniques is mainly reflected in the
following aspects [39]:
(1) The coordinate origin is inconsistent. For example, the coordin ate origin of VTRF (VLBI
Terrestrial Reference Frame) is set by taking one SLR station’s geocentric coordinate defined
relative to the Earth’s CM, and the station is co-located with VLBI and SLR, whose geocentric
coordinate error will cause the deviation of the coordinate origin. In addition, different analysis
centres may take different SLR stations to define the VTRF’s coordinate origin, which will lead
to deviation of the origin between each VTRF and between the VTRF and other terrestrial
reference frames. Although the terrestrial reference frames realized by SLR, LLR (lunar laser
ranging),GPS and other space techniques are achieved by dynamic techniques, the dynamic
reference systems adopted by various techniques are different, and therefore, the coordinate
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origin implemented by them may not be exactly the same. Besides, various dynamic techniques
only determine the coordinate origin of the reference frame relative to the Earth’s CM to a
certain precision.
(2) Scale is not entirely consistent. The scale o f each terrestrial reference frame achieved by a
variety of space techniques is jointly determined by the value of the speed of light C, the Earth’s
gravitational constant GM(G is a gravitational constant, M represents the mass of the Earth)
and the relativistic correction model. If each analysis centre uses different constants and models,
thiswill lead to different scales for each terrestrial reference frame. Even if each analysis centre
uses exactly the same constant and model, different observation techn iques, different amounts
of data and data processing methods also lead to a slightly different scale.
(3) Orientation differs. The defining axis point of the terrestrial reference frame is inconsistent
when using different epochs and the Earth’s rotation parameters of different systems;there is a
slight rotation between them. For example, although the BTS (Bureau International de
l'Heure(BIH) Terrestrial System) andITRF sequence used BIH orientation parameters at epoch
1984.0, the other reference frame has adopted each defined parameter.
For two different frames, there are many transformation models when converting to the
same frame. Here, we adopt the widely-used seven-parameter transformation and Hermelt
transformation [23,40–43]. The transformation between two frames is as follows:
X2 = X1 + T + DX1 + RX1
(1)
where T (T1, T2, T3 ) are three translation parameters representing the difference of origins;
R (R1, R2, R3 ) are three rotation parameters denoting the difference in orientation; D is the scale
deviation. X 1 and X 2 are two vectors representing two different coordinates of one co-located site
based on different reference frames obtained by two different observation techniques.
Due to the effect of plate motion, regional crustal deformation, postglacial rebound, and so on ,
station coordinates of the terrestrial reference frame will change over time. In order to maintain the
stability of the conventional terrestrial reference frame, a high -precision reference frame always
corresponds to a time epoch and a velocity field. The velocity field can be determined by a certain
plate motion model (e.g., NNR-NUV EL1A [30,31] or APKIM2005 (ActualPlate Kinematic and
crustal deformation Model) [44]) and can also be derived by observation results; nowadays a
combination of the two is preferred. When the epoch of the conventional terrestrial reference frame
changes, the transformation parameters will change,as well. Suppose that the rate of change of
transformation parameters between two r eference frames is 𝑇 𝑇1 , 𝑇2 , 𝑇3 , 𝑅 𝑅1 , 𝑅 2 , 𝑅3 and 𝐷 . Then,
Equation (1) is derived by time t, and Equation (2) is obtained as follows:
X2 = X 1 + T + D X1 + DX1 + RX1 + RX1
(2)
Because the rotation matrix is typically less than 10 milliseconds, when transformed to radians,
its magnitude is 10 −7–10 −8, and the order of the velocity field 𝑋1 is just a few centimetres per year, so
the item 𝑅𝑋1 in Equation (2 ) is relatively small and can be neglected. The scale parameter D is
generally in the magnitude of 10 −6–10 −7, so it can be neglected,as well. Thus, Equation (2) can be
simplified as follows:
X2 = X 1 + T + D X1 + RX1
(3)
The expanded Equation (3) is:
𝑋
Y
Z
2
𝑋
= Y
Z
𝑇1
𝑋
+ T2 + 𝐷 Y
Z
T3
1
+
1
0
𝑅3
− 𝑅2
−𝑅3
0
𝑅1
𝑅2
−𝑅1
0
𝑋
Y
Z
(4)
1
where X1 and X 2 represent two velocity fields based on different reference frames, respectively.
The vertical movement surveyed by SLR is relative to the Earth’s CM. This means that there is a
systematic bias between other velocity fields and the SLR velocity field. In order to obtain the
systematic bias, co-located stations should be prepared using two techniques. Because different
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reference frames are adopted, the observation values of co-located stations are different. After
systematic bias has been corrected, different velocity fields can be unified to a m ovement relative to
the Earth’s CM. Systematic bias can be solved in an iterative manner, but the premise is to use the
co-located data in two different reference frames. By eliminating the systematic bias in the geocentric
reference frame and then minimizing the sum of the square of the projection of the difference
between each vertical velocity and the SLR value on every co-located site in the geocentric direction,
the systematic bias can be calculated by iteration using the following equation [19,45]:
𝑚
𝑉𝑖1 − 𝑆 − 𝑉𝑖2 ∙ 𝑋𝑖
2
= 𝑚𝑖𝑛
(5)
𝑖 =1
where 𝑉𝑖1 and 𝑉𝑖2 are the projection vectors of the other technique’s vertical velocity field and the
SLR’s one on the geocentric reference frame, respectively, 𝑆 is the systematic bias, 𝑋𝑖 is the unit
geocentric vector of the i-th station and m is the number of stations.
If the velocity field is provided with site velocity in the geocentric reference frame, it can be
transformed into a topocentric coordinate system along the directions of east, north and vertical
using Equation (6):
𝑉𝑖𝑒
− sin 𝜆 𝑖
𝑛
−
sin 𝜑𝑖
𝑉𝑖 =
𝑢
cos 𝜑𝑖 cos 𝜆 𝑖
𝑉𝑖
cos 𝜆 𝑖
− sin 𝜑𝑖 sin 𝜆 𝑖
cos 𝜑𝑖 sin 𝜆 𝑖
0
cos 𝜑𝑖
sin 𝜑𝑖
𝑉𝑖𝑥
𝑦
𝑉𝑖
𝑉𝑖𝑧
(6)
where 𝜑𝑖 and 𝜆𝑖 are the geocentric latitude and longitude of the i-th station, respectively, 𝑉𝑖𝑥 ,
𝑦
𝑉𝑖 and 𝑉𝑖𝑧 are the three components of the geocentric velocity of the i-th station, respectively, and 𝑉𝑖𝑒 ,
𝑉𝑖𝑛 and 𝑉𝑖𝑢 are those of the topocentric velocity, respectively.
In Equation (5), the projection vectors of the velocity field’s vertical component on the
geocentric reference frame are used in a systematic bias model. Therefore, vertical vectors should be
calculated relative to the geocentric reference frame. The equation is as follows:
𝑣𝑖𝑥
cos 𝜑𝑖 cos 𝜆 𝑖 𝑉𝑖𝑢
𝑦
𝑣𝑖 = cos 𝜑𝑖 sin 𝜆 𝑖 𝑉𝑖𝑢
sin 𝜑𝑖 𝑉𝑖𝑢
𝑣𝑖𝑧
In Equation (4), 𝑇 𝑇1 , 𝑇2 , 𝑇3 ,𝑅 𝑅 1 , 𝑅 2 , 𝑅3
(7)
and 𝐷 denote the rate of change of systematic bias
between two velocity fields based on different reference frames, while 𝑆 in Equation (5 ) represents
the systematic bias of two vertical velocity fields. Therefore, the systematic bias and the rate of
change of two vertical velocity fields can be resolved from the combination of Equations (4) and (7).
If X 1 represents the vertical velocity field obtained by the other t echnique, while X 2 is that
obtained by SLR, the two vertical velocity fields can be imputed to the geocentric reference frame by
adopting Equations (4) and (7). Ignoring the impact of the rates of change of the rotation and
translation parameters, the error equation can be given as follows:
𝑉1
1
𝑉2 = 0
𝑉3
0
0 0 0
1 0 𝑍
0 1 −𝑌
−𝑍
0
𝑋
𝑇1
𝑇2
𝐿1
𝑌 𝑋 𝑇3
−𝑋 𝑌 𝑅 1 + 𝐿2
𝐿3
0 𝑍 𝑅
2
𝑅3
𝐷
(8)
where X, Y and Z are the Cartesian coordinates of the co-located station obtained by two different
techniques and matrix L represents the difference between the projections of two vertical velocities
in two different reference frames, as follows:
𝐿1
𝑋
𝐿2 = Y
𝐿3
Z
1
𝑋
− Y
Z
(9)
2
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In addition, in order to analyse the difference between the SLR vertical reference frame and
the other vertical velocity field, the fitting slopes of both results are calculated by the least squares
linear fitting method using Equation (10), and the correlation coefficients are compared using
Equation (11) [19,45]:
y =𝑎∙𝑥+𝑏
ρ𝑥𝑦 =
𝑐𝑜𝑣 𝑥, 𝑦
𝐷 𝑥
𝐷𝑦
(10)
(11)
wherex and y are the vertical velocities obtained by different space techniques, respectively, a and b
are the fitting coefficientsand ρ𝑥𝑦 is the correlation coefficient of the two results.
By using the fitting function and correlation, it is possible to analyse not only the difference
between the vertical velocities obtained by SLR and the other technique, but also the consistency
between velocity fields after correction by the systematic model, and then , by analysis and
discussion, the reasonableness of taking the SLR vertical velocity as the reference frame of vertical
crustal movement can be determined.
3. Data
The data used herein come from the ITR F, including the SLR, VLBI and GPS site coordinates
and velocity fields in the ITRF2005 and ITRF2008 series, whose reference epochs are 2000.0 and
2005.0, respectively, solved by each data analysis centre. The following data sequence can be found
on the official website of ITR F [46].
The SLR velocity field is chosen from the ITRF2008_SLR coordinate results submitted by each
analysis centre. The solution deals with data from 1983.0–2009.0, comprising 26 years ofobservation
data. The precision of station velocities is mostly better than 3 mm/yr.
The VLBI velocity field comes from ITR F2008_VLBI and ITRF2005_VLBI. The time sequence of
ITRF2008_VLBI is from 1980.0–2009.0, comprising 29 years ofobservation data made up of the
geocentric coordinates and velocities of VLBI stations. The reference epoch is 2005.0. The nominal
precision of the geocentric coordinate is 3 mm. The precision of most geocentric coordinate velocities
is better than 3 mm/yr. ITRF2005_VLBI.SCC adopted the 1992.9–2005 time series, a data span of nine
years. The precision of most geocentric coordinate velocities is better than 3 mm/yr.
The GPS velocity field is from ITR F2008_GNSS and ITRF2005_GPS. The time span of
ITRF2008_GNSS.SCC is from 1997.0–2009.5, comprising 12.5 years. The nominal accuracy of the
geocentric coordinate obtained is better than 1 mm, and the station velocity accuracy is less than
1 mm/yr. ITRF2005_GPS.SCC contains GPS phase observation data from 1996.0 –2006.0, and the
nominal accuracy of most of its geocentric coordinates is better than 1 mm, while the accuracy of the
stations’ horizontal velocities is mostly less than 0.2 mm/yr, and the vertical accuracy is better than
1 mm/yr.
Information about co-located sites was supplied by IERS (International Earth Rotation and
reference systems Service) at01 April 2010.
During data preprocessing, the following aspects are mainly considered: (1) whether the
variance of station velocity of each component is better than 3 mm/yr; (2) whether the station is
located in the border zone of the deformation plate or the internal plate’s apparent tectonic
deformation zone; (3) whether the three components of the co-located site’s velocity remain in the
same direction and of the same order of magnitude. By comprehensive application of the above
criteria, the precision and stability of selected stations and co-located observations can be
initially determined.
4. Resultsand Analysis
According to the above methods and observation data, the co-located sites (numbers of
co-location sites shown in Tables 2, 4, 6 and 8) of SLR , GPS and VLBI in different reference frames
were selected, and then, the definition of the reference frame of global vertical crustal movements
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was studied by using the SLR vertical velocity in ITRF2008, which we defined as the SLR vertical
reference frame (SVRF). The systematic bias between the SVRF and vertical velocities of GPS and
VLBI, respectively obtained by using Equations (4), (5) and (7), namely the rate of change of
transformation between them, is shown in Tables 1, 3, 5 and 7.
In order to analyse the difference between SVRF and another vertical velocity, the method of
linear regression analysis is adopted to resolve the regression coefficients between them. The fitting
slopes and regression coefficients between the vertical velocities of SVRF, GPS and VLBI can be
solved by using Equations (10) and (11), and the results are shown in Tables 2, 4, 6 and 8, along with
the fitting slope (a, b) and the regression coefficients between SVRF and the other velocity field
before and after correction.
Table 1. The rates of change of transformation parame te rs:ITRF2008_GNSS/SLR ve rtical re fere nce
frame (SVRF).
Translations (mm/yr)
Rotations (mas/yr)
Scale (10−9 /yr)
T1
T2
T3
R 1
R 2
R 3
D
0.0473
−0.1017
0.1027
−0.0020
−0.0002
0.0004
0.0710
Table 2. Fitting slope and corre lation coe fficie nt:SLR ve rtical re fere nce frame (SVRF)/ITRF2008_GNSS.
Data Status
Be fore Correcte d
Afte r Correcte d
a
0.5740
0.5672
b
0.5559
0.2095
Correlation Coefficient
0.6404
0.7172
No.of Co-Location
29
Table 3. The rates of change of transformation parame ters:ITRF2008_VLBI / SVRF.
Translations (mm/yr)
Rotations (mas/yr)
Scale (10−9 /yr)
T1
T2
T3
R 1
R 2
R 3
D
0.0065
−0.1758
0.1663
−0.0013
0.0006
0.0006
−0.0376
Table 4. Fitting slope and corre lation coe fficie nt:SVRF / ITRF2008_VLBI.
Data Status
a
b
Correlation Coefficient
Be fore Correcte d
0.7062
−0.3142
0.7406
Afte r Correcte d
0.6953
−0.1381
0.7650
No.of Co-Location
11
Table 5. The rates of change of transformation parame ters :ITRF2005_GPS / SVRF.
Translations (mm/yr)
Rotations (mas/yr)
Scale (10−9 /yr)
T1
T2
T3
R 1
R 2
R 3
D
0.2933
-0.1661
0.2628
−0.0030
−0.0039
0.0001
0.0339
Table 6. Fitting slope and corre lation coe fficie nt:SVRF / ITRF2005_GPS.
Data Status
a
b
Correlation Coefficient
Be fore Correcte d
0.9586
0.1143
0.8545
Afte r Correcte d
1.0117
0.3402
0.8602
No.of Co-Location
22
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Table 7. The rates of change of transformation parame ters :ITRF2005_VLBI / SVRF.
Translations (mm/yr)
Rotations (mas/yr)
Scale (10−9 /yr)
T1
T2
T3
R 1
R 2
R 3
D
1.5500
1.0856
0.8700
0.0198
−0.0299
0.0039
0.2845
Table 8. Fitting slope and corre lation coe fficie nt:SVRF / ITRF2005_VLBI.
Data Status
a
b
Correlation Coefficient
Be fore Correcte d
0.8640
−0.2135
0.8976
Afte r Correcte d
1.2805
1.2598
0.9352
No.of Co-Location
9
From the above Tables about the rates of change of transformation parameters , it can be seen
that the systematic bias between the SVRF, VLBI and GPS velocities is very small, and the rate of
change of transformation parameters is almost focused in the order of millimetres per year; the
rotation parameters are milliarcseconds;and the scale parameter is smaller,as well: basically in the
order of 10 −11. Overall, the above obtained systematic biases are different in size and direction, which
indicates that the reference frame of vertical crustal movement indeed does not unify the vertical
velocities given by each data analysis centre. Comparing the systematic differences of Tables 1 and 5,
it can be shown that the reference datum of vertical crustal movements differs even with the same
observation techniques at different observation epochs and time spans, and the same analysis results
are obtained by comparing Table 3 to Table 7.
From the above Tables about the fitting slope and correlation coefficient , the correlation
coefficient has been improved after systematic bias correction; that is to say, the fitting degree
between the SVRF and the vertical velocity of each technique after correction is better than before. In
addition, we give an example to show the difference between them. Table 9 shows the comparison
between SVRF and the velocity of ITRF2008_GNSS, as well as those corrections after systematic bias,
respectively. Figure 1 shows the difference between the ITRF2008_GNSS vertical-velocity field and
SVRF before and after systematic bias correction, where the left side is the difference between the
SVRF and ITRF2008_GNSS velocity field, and the right is the difference after systematic bias
correction in the corresponding representation. It can be seen from the above results that after the
correction of systematic bias in the whole station, the consistency between SLR and the
corresponding station’s vertical velocities is getting better. While the con sistency of som e stations
isgetting worse, which indicates that, on the one hand, a difference between SLR and the other
technique really exists, on the other hand, the process of solving systematic bias may also bring
certain rounding errors.
Figure 1. Diffe re nces be twee n the ITRF2008_GNSS vertical-ve locity fie ld and the SLR ve rtical
re fe re nce frame (SVRF) be fore (blue ) and afte r (ye llow) systematic bias correction.
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Table 9. Ve locity comparison: ITRF2008_GNSS/SLR vertical re fe rence frame (SVRF).
Numbering
SiteName
Latitude
(°)
Longitude
Vu(mm/yr)
Vu(mm/yr)
Vu(mm/yr)
(°)
before Corrected
after Corrected
SVRF
1
GRAS
43.7547
6.9205
0.9497
0.3984
−0.5234
2
GRAZ
47.0671
15.4934
0.9755
0.4328
0.8417
3
BOR1
52.2769
17.0734
−0.1985
−0.3555
−0.3553
4
CRAO
44.4132
33.9909
0.9494
0.4342
−0.8121
5
CAGL
39.1359
8.9727
0.6292
0.0828
0.0828
6
MATE
40.6491
16.7044
1.2091
0.6748
0.6317
7
HERS
50.8673
0.3362
−0.4892
−0.0754
−0.3211
8
SFER
36.4643
−6.2056
0.8938
0.3301
0.1145
9
ZIMM
46.8770
7.4652
1.8407
1.2892
1.3675
10
WAB2
46.9237
7.4642
1.4977
0.9459
1.3675
11
POTS
52.3792
13.0660
−0.3229
−0.2309
−0.2309
12
WTZR
49.1441
12.8789
−0.4793
−0.0707
1.1585
13
BJFS
39.6086
115.8924
2.6381
2.2052
2.9482
14
SHAO
31.0996
121.2004
1.2936
0.8806
−1.4123
15
KGNI
35.7103
139.4881
1.8650
1.4328
2.1445
16
MTKA
35.6795
139.5613
0.5400
0.1011
2.1445
17
HRAO
−25.8901
27.6869
−0.2437
−0.2067
1.1900
18
SUTH
−32.3802
20.8104
0.5283
0.1005
0.0988
19
ALGO
45.9558
−78.0713
4.1402
3.5374
3.3044
20
QUIN
39.9745
−120.9444
1.6741
1.1031
−0.5715
21
MDO1
30.6805
−104.0149
1.6503
1.0653
−0.2696
22
MAUI
20.7066
−156.2570
−0.6920
−1.1620
−1.1627
23
GODE
39.0217
−76.8268
−0.5481
−1.1447
−1.1446
24
MONP
32.8919
−116.4223
1.3620
0.7885
0.9013
25
AREQ
−16.4655
−71.4927
−0.9889
−1.5005
−1.8695
26
TIDB
−35.3992
148.9799
0.8747
0.5512
0.9016
27
YAR1
−29.0465
115.3469
0.3635
0.0571
−0.1080
28
STR2
−35.3161
149.0101
1.1871
0.8651
0.9016
29
THTI
−17.5770
−149.6064
−0.4598
−0.0133
0.0382
of Sites
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The reference frame’s origin established by the SLR technique is defined relative to the Earth
CM by taking the three first-order coefficients of the Earth’s gravitational field model as zero, so the
observation of SLR is truly with respect to the Earth CM. From the basic principles of geodetic VLBI,
VLBI is a pure geometric measurement, which is not sensitive to the Earth’s CM. In establishing the
VLBI reference frame, the origin is typically determined through the co-location of the SLR station’s
coordinates and relative baseline vector; that is to say, the reference frame-based VLBI technique is
not truly relative to the Earth’s CM. Although high-precision GPS is a relative measurement, it is
also solved by taking the Earth’s gravitational field’s three first -order coefficients as zero to ensure
that the station coordinates are relative to the centre of the Earth. Therefore, in another sense, the
reference frame established by the GPS technique is relative to the centre of the Earth, but it is not as
direct as SLR; and the accuracy of the geocentric position obtained from GPS is not as good as SLR
either [38]. Compared tothe SLR technique, the deficiencies in solar radiation pressure modelling of
GNSS techniques affect the GNSS-derived geocenter series, especially the Z component [36,47].
Otherwise, the uncalibrated satellite antenna phase centre offsets of GNSS techniques are a major
error source for the scale determination of the reference frame [36,48].
In summary, when taking the SLR vertical movement as the global reference frame of vertical
crustal motion, the systematic bias between the SVRF and other velocity fields is relatively small.
SLR is one of the most accurate space techniques for monitoring the geocentric motion, and it can
directly measure the ground station’s geocentric coordinate and velocity relative to the centre of the
Earth’s mass. Therefore,it is feasible to take the vertical velocity of the SLR technique obtained as a
reference frame of vertical crustal motion for study, as it has the most direct geophysical significance
and geometric meaning. Furthermore, with the progress of the SLR space observation technique, the
precision and number of observations are constantly improving, and simultaneously , the station’s
velocity field in the framework of ITRF2008 is based on the time series, so it is currently a reasonable
and feasible implementation to take the SLR vertical velocity field as the unified global reference
frame of vertical crustal movement.
5. Conclusions
Vertical crustal movement affects people’s living environment; therefore, it is of far -reaching
importance to study and unify its reference frame on a global scale. This paper started from the
feasibility of establishing a unified global vertical reference frame for crustal motion and then
analysed and discussed the proposed method of taking the SLR vertical movement as the vertical
reference frame, which was verified by the corresponding measured data. The conclusions are
as follows:
The SLR technique is carried out by observing the distance from the ground station to the
satellite to obtain the station’s location with respect to the centre of the Earth’s mass; namely, the
ground station’s geocentric coordinate can be directly measured by the SLR technique, and thus, the
station’s vertical velocity obtained by the SLR technique is relative to the centre of the Earth’s mass.
Therefore,it is feasible to define the SLR vertical velocity field in the framework of ITRF2008 as the
global reference frame of vertical crustal movement, which contains more extensive geophysical
information, has a developed and improved characteristics, is directly relative to the centre of the
Earth’s mass, and so on . In addition, the origin of the ITR F2008 framework is defined by usi ng SLR
data, so it is convenient and easier to unify the station’s coordinate frame and the vertical
movement’s framework. Furthermore, Crespi et al. [26] underlined the unbreakable link between
geodesy and geophysics; that is to say, geodesy requires the b est models to explain geophysical
phenomena, while geophysical models require continuous analysis of space geodetic observations.
The SLR vertical reference frame is merged with both the geophysical meanings and geodetic
observations, and thus, here, we recommend taking the SLR vertical velocity under ITRF2008 as the
global reference frame of vertical crustal movement. In this paper, it is verified as the reference frame
by using ITRF velocity field data. Because ITRF has already made internal integration of frames
when implemented [6], the systematic bias solved here is smaller. In future work, we will use other
forms of data to check whether the above method is reasonable or not, such as data from analysis
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centres on the world and solved in each of their fr ameworks. There is a referenced value, and the
significance for the establishment of a unified global or regional reference frame of vertical crustal
movements by the above systematically expounded related methods.
Ackno wledgments:The authors grate fully acknowle dge IERS and ITRF for providing data and products. This
study was supporte d by the National Natural Scie nce Foundation of China (Grant No. 41374027).
Author Contributions: Xinhui Zhu and Fuping Sun provide d the initial idea for this research. Xinhui Zhu
wrote the pape r and analyse d the data and results. Re n Wang conce ive d of and designe d theexpe rime nts and
pe rforme d the data handling. Jinling Wang contribute d to the structure and gave valuable advice on the pape r
writing. All authors contribute d to the production of this publication.
Conflicts of Interest: The authors declare no conflict of interest.
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